| L(s) = 1 | + 4·7-s − 2·25-s − 20·31-s + 18·49-s − 28·73-s + 40·79-s − 4·97-s − 56·103-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s − 8·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 2/5·25-s − 3.59·31-s + 18/7·49-s − 3.27·73-s + 4.50·79-s − 0.406·97-s − 5.51·103-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s − 0.604·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.382623844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382623844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 94 T^{2} + 6027 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24149046544847600839073552901, −7.03769540968738111850309825999, −6.94679222661206738265920979859, −6.75413502859813980266114784138, −6.41643064417248935499572875009, −6.07180940167385785464572069016, −5.69953147403795672436086624383, −5.66263481165500652538892055716, −5.45383661855659355418933396117, −5.31412908788555003760873862525, −4.92901224701463792521965998459, −4.77181274146763543101832909328, −4.38143671440090286964158824815, −4.20998673994309506133722944488, −3.89893133420175609926305559963, −3.74469558901146948320935472133, −3.38068711076203999920707988420, −3.20411459514129297215561107937, −2.60062037510907051134444169126, −2.30862903333473820109330197806, −2.24521132614957376419970949461, −1.71039540932488136867696777869, −1.36934781319433287747089471348, −1.21941888614437271042163266674, −0.27759676442585793268376754337,
0.27759676442585793268376754337, 1.21941888614437271042163266674, 1.36934781319433287747089471348, 1.71039540932488136867696777869, 2.24521132614957376419970949461, 2.30862903333473820109330197806, 2.60062037510907051134444169126, 3.20411459514129297215561107937, 3.38068711076203999920707988420, 3.74469558901146948320935472133, 3.89893133420175609926305559963, 4.20998673994309506133722944488, 4.38143671440090286964158824815, 4.77181274146763543101832909328, 4.92901224701463792521965998459, 5.31412908788555003760873862525, 5.45383661855659355418933396117, 5.66263481165500652538892055716, 5.69953147403795672436086624383, 6.07180940167385785464572069016, 6.41643064417248935499572875009, 6.75413502859813980266114784138, 6.94679222661206738265920979859, 7.03769540968738111850309825999, 7.24149046544847600839073552901
